Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

A pair of Sudoku puzzles that together lead to a complete solution.

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

Four small numbers give the clue to the contents of the four surrounding cells.

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

You need to find the values of the stars before you can apply normal Sudoku rules.

A challenging activity focusing on finding all possible ways of stacking rods.

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

A few extra challenges set by some young NRICH members.

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

Use the differences to find the solution to this Sudoku.

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

This Sudoku, based on differences. Using the one clue number can you find the solution?

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

Can you use your powers of logic and deduction to work out the missing information in these sporty situations?

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

A man has 5 coins in his pocket. Given the clues, can you work out what the coins are?

Find the values of the nine letters in the sum: FOOT + BALL = GAME

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

Two sudokus in one. Challenge yourself to make the necessary connections.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?